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| Mirrors > Home > MPE Home > Th. List > cnconst | Structured version Visualization version GIF version | ||
| Description: A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.) |
| Ref | Expression |
|---|---|
| cnconst | β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ (π΅ β π β§ πΉ:πβΆ{π΅})) β πΉ β (π½ Cn πΎ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fconst2g 7199 | . . . 4 β’ (π΅ β π β (πΉ:πβΆ{π΅} β πΉ = (π Γ {π΅}))) | |
| 2 | 1 | adantl 481 | . . 3 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (πΉ:πβΆ{π΅} β πΉ = (π Γ {π΅}))) |
| 3 | cnconst2 23137 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ) β§ π΅ β π) β (π Γ {π΅}) β (π½ Cn πΎ)) | |
| 4 | 3 | 3expa 1115 | . . . 4 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (π Γ {π΅}) β (π½ Cn πΎ)) |
| 5 | eleq1 2815 | . . . 4 β’ (πΉ = (π Γ {π΅}) β (πΉ β (π½ Cn πΎ) β (π Γ {π΅}) β (π½ Cn πΎ))) | |
| 6 | 4, 5 | syl5ibrcom 246 | . . 3 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (πΉ = (π Γ {π΅}) β πΉ β (π½ Cn πΎ))) |
| 7 | 2, 6 | sylbid 239 | . 2 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (πΉ:πβΆ{π΅} β πΉ β (π½ Cn πΎ))) |
| 8 | 7 | impr 454 | 1 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ (π΅ β π β§ πΉ:πβΆ{π΅})) β πΉ β (π½ Cn πΎ)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 {csn 4623 Γ cxp 5667 βΆwf 6532 βcfv 6536 (class class class)co 7404 TopOnctopon 22762 Cn ccn 23078 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-topgen 17395 df-top 22746 df-topon 22763 df-cn 23081 df-cnp 23082 |
| This theorem is referenced by: xrge0mulc1cn 33450 cxpcncf2 45169 |
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